Direct Simulations of Mixing of Liquids with Density and ViscosityDifferencesJ. J. Derksen*

Chemical & Materials Engineering, University of Alberta, Edmonton, Alberta, T6G 2G6 Canada

ABSTRACT: Simulations of flow and scalar transport in stirred tanks operated in transitional and mildly turbulent regimes (Re= 3000−12000) are presented. The moderate Reynolds numbers allow the flow to be simulated directly, without the use ofturbulence closure or subgrid-scale models. The Newtonian liquids that are blended have different densities and/or viscosities,and the emphasis is on how these differences affect mixing times. The density difference is characterized by a Richardson number(Ri) that varies in the range of 0−0.5. The kinematic viscosity ratio is between 1 and 4. The results show that mixing timesincrease steeply with increasing Ri and that changing the tank layout can partly mitigate this effect. The viscosity ratio has a muchweaker influence on the mixing time.

■ INTRODUCTIONMixing of miscible liquids (i.e., blending) is a basic operation inmany food, pharmaceutical, and chemical processes. For batchprocesses, mixing is usually carried out in stirred tanks. Thetime to homogenization (mixing time) is a key processcharacteristic that depends on the geometry of the mixingdevice (impeller and tank layout), the operating conditions(such as impeller speed), and the liquids involved in theprocess. If the liquids involved all have the same uniformmechanical properties (i.e., they all have the same density andviscosity and are Newtonian), the dimensionless mixing time(e.g., represented by a number of impeller revolutions until acertain predefined level of homogeneity is reached) depends onthe geometry as defined in geometrical aspect ratios and theReynolds number (and possibly the Froude number if the tankhas a free surface). Uniformity of the mechanical properties ofthe tank’s contents before the blending process is completed,however, is not a common situation; the liquids to be blendedusually have different viscosities and/or densities. It is expectedthat the density and viscosity differences have consequences forthe flow structures in the tank and, therefore, for the mixingtime.With nonuniform contents in the agitated tank, the number

of (nondimensional) parameters characterizing the mixingprocess quickly grows: Not only do multiple densities andviscosities enter the parameter space, but the initial conditionsand/or the way in which the various liquid components areadded to the tank also come into play. Investigating mixingtimes over the entire parameter space and for all blendingscenarios is obviously impossible. It is the intent of this articleto focus on the consequences of density and viscositydifferences on mixing times for a limited number of situationsinvolving two different liquids that are initially completelysegregated and each occupy roughly one-half of the tankvolume. The latter implies that the mechanical properties ofboth liquids determine the overall flow in the tank; as anexample, we do not consider small additions of liquid A in alarge pool of liquid B. In the latter situation, the properties ofone liquid (i.e., B) likely dominate the mixing process.

The specific systems that are considered are conceptuallysimple. The two miscible liquids are placed in a mixing tankwith the interface between them at half of the tank height. If thetwo liquids have different densities, the light liquid occupies theupper part of the volume, and the heavy liquid occupies thelower part. Until time zero, this fully segregated system has zerovelocity everywhere. Then, at time zero, the impeller isswitched on to a constant angular velocity, and the blendingprocess is monitored. The impeller used throughout thisresearch is a pitched-blade turbine with four blades under anangle of 45°; the tank is cylindrical and has a flat bottom andfour equally spaced baffles along its perimeter. In a recentarticle of ours,1 the above scenario was executed to assess theeffect of density differences on mixing time. The scope of thepresent article is much broader than that of ref 1 in that, now,viscosity differences and also different stirred-tank layouts areconsidered, in addition to density differences.The research described here is purely computational with an

interest in turbulent mixing. As is well-known, for simulations ofstrongly turbulent flows, turbulence modeling is required;direct simulation of the flow is generally not an option becauseof the broad spectrum of length scales involved2 that would allneed to be resolved. In turbulence modeling, parametrizationsfor the smaller length scales are applied (as in large-eddysimulation),3 or the statistics of the fluctuations present in theflow are modeled and taken into account in the (ensemble- ortime-averaged) equations of fluid motion (Reynolds averag-ing).4 Single-phase, turbulent flow in complex geometries suchas agitated tanks is a clear challenge for turbulence models.5−9

This is because the localized energy input (the stirrer) inducesvery strong inhomogeneity and anisotropy and keeps theturbulence away from a dynamic equilibrium state. To thiscomplexity, we now add a liquid composition field that varies inspace and time, with the viscosity and density being a function

Received: January 5, 2012Revised: April 3, 2012Accepted: April 22, 2012Published: April 23, 2012

Article

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of the local, time-dependent composition. Because our focus ison the effects of the local viscosity and density on the mixingprocess, we do not want our results to be (potentially)corrupted by the way turbulence models deal with theselocalized effects. For this reason, we do not apply turbulencemodeling in our simulations. The price to pay is that we arelimited to relatively modest Reynolds numbers (in this work,the default value is Re = 6000 with some excursions toward12000, where Re is precisely defined below), so that we (atleast) strive for direct resolution of the flow. The choice fordirect simulations requires a study of the effects of gridresolution on the flow field and on the mixing time results.This article is organized in the following manner: First, the

flow system is described, dimensionless numbers are defined,and the parameter ranges covered in this research are identified.Subsequently, the simulation procedure is outlined schemati-cally with references to the literature for further details. Wethen present results. The emphasis is on the effect of a densitydifference on the mixing time and on ways to mitigate theprohibitively long mixing times encountered in some cases bychanging the impeller location and/or direction of rotation.Because liquids with different densities often also have differentviscosities, flow with local viscosity variation was studied aswell. The impact of grid resolution is discussed in a separatesubsection. Conclusions are summarized in the last section.

■ FLOW SYSTEMSThe tank and agitator and the coordinate system used in thiswork are shown in Figure 1. The tank is cylindrical, with four

equally spaced baffles along its perimeter. The flow is driven byfour pitched (45°) blades attached to a hub that is mounted ona shaft that runs over the entire height of the tank. The tank isclosed off with a lid so that, at the top surface (as on all othersolid surfaces), no-slip conditions apply. The Reynolds numberof this flow system is defined as Re = (ND2/ν ̅), where N is theimpeller speed (in revolutions per second), D is the impellerdiameter (see Figure 1), and ν̅ is the tank-volume-averagedkinematic viscosity. Figure 1 shows the geometry with theimpeller placed at a distance of C = T/3 above the flat tankbottom. We studied the consequences of changing the off-bottom clearance by also performing simulations for C = T/2.

The impeller was operated in both down-pumping and up-pumping modes.Initially, two layers of liquid are placed in the tank, with their

interface at z = 0.5H. The upper liquid has a density that is Δρless than that of the lower liquid; that is, the system starts froma stable stratification. The denser liquid has either the samekinematic viscosity as the lighter liquid or a higher viscosity (νD≥ νL, where νD and νL are the viscosities of the dense and lightliquids, respectively). Starting from a completely still situation,we switch on the impeller with constant speed of N.In addition to the Reynolds number, a Richardson number

defined as Ri = (gΔρ/ρ̅N2D) and the kinematic viscosity ratioVi ≡ νD/νL fully pin down the flow system. In the expressionfor Ri, g is gravitational acceleration, and ρ̅ is the volume-averaged density of the liquid in the tank. Rielly and Pandit10

defined the Richardson number as gΔρH/ρN2D). Given the(standard) aspect ratios used in the present work, the latterexpression is equal to 3 times the Richardson number asdefined herein.The default Reynolds number used was 6000. Systems with

Re = 3000 and 12000 were simulated as well. Specifically, forthe latter value, resolution effects need to be carefully assessed.This Reynolds number range covers transitional and mildlyturbulent flowat least for single-liquid systems. TheRichardson number ranged from 0.0 to 0.5, and the viscosityratio from 1.0 to 4.0. The latter are moderate viscosity ratios, sothat very thin striations of the highly viscous phase in the lessviscous phase11 are not to be expected. As for the higherReynolds number, the impact of spatial resolution on theblending process was assessed for the highest viscosity ratio, Vi= 4.0.

■ MODELING APPROACHThe lattice-Boltzmann method (LBM) was applied to numeri-cally solve the incompressible flow equations.12,13 The methodoriginates from the lattice-gas automaton concept as conceivedby Frisch, Hasslacher, and Pomeau in 1986.14 Lattice gases andlattice-Boltzmann fluids can be viewed as collections of(fictitious) fluid particles moving over a regular lattice andinteracting with one another at lattice sites. These interactions(collisions) give rise to viscous behavior of the fluid, just ascollisions/interactions among molecules do in real fluids. Themain reasons for employing the LBM for fluid flow simulationsare its computational efficiency and its inherent parallelism,both not being hampered by geometrical complexity.In this article, the LBM formulation of Somers15 is employed.

It falls in the category of three-dimensional, 18-speed (D3Q18)models. Its grid is uniform and cubic. Planar, no-slip wallsnaturally follow when bounce-back conditions are applied. Fornonplanar and/or moving walls (which we have because we aresimulating the flow in a cylindrical, baffled mixing tank with arevolving impeller), an adaptive force field technique (i.e., animmersed boundary method) has been used.16,17

The local composition of the liquid is represented by a scalarfield c (with c = 1 representing pure light fluid and c = 0representing pure dense fluid) for which we solve the transportequation

∂∂

+ ∂∂

= Γ ∂∂

ct

ucx

cxi

i i

2

2(1)

(with summation over repeated indices) where ui is the ithcomponent of the fluid velocity vector and Γ is a diffusion

Figure 1. Stirred-tank geometry considered: a baffled tank with apitched-blade impeller. The coordinate systems [(r,z) and (x,y,z)] arefixed and have their origin at the center of the bottom of the tank. Thetop of the tank is closed off with a lid. In addition to an impeller-to-bottom distance of C = T/3 (as shown in the figure), a distance of C =T/2 was considered as well. The impeller can rotate both ways, so thatit can pump downward (default) and upward.

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coefficient that follows from setting the Schmidt number, Sc ≡ν/Γ, equal to 1000. We solve eq 1 with an explicit finite-volumediscretization on the same (uniform and cubic) grid as theLBM. A clear advantage of employing a finite-volumeformulation is the availability of methods for suppressingnumerical diffusion. As in previous works,18,19 total-variation-diminishing (TVD) discretization with the Superbee flux limiterfor the convective fluxes20 was employed. We step in timeaccording to an Euler explicit scheme. This explicit finite-volume formulation for scalar transport does not hamper theparallelism of the overall numerical approach.Strictly speaking, the Schmidt number is the fourth

dimensionless number (next to Re, Ri, and Vi) defining theflow. Its large value (103) makes the micro-scalar scale(Batchelor scale) a factor of (Sc)1/2 ≈ 30 smaller than theKolmogorov length scale and quite impossible to resolve in our

numerical simulations. In the simulationsalthough wesuppress numerical diffusion as much as possiblediffusion iscontrolled by the grid spacing, and the precise value of Sc basedon molecular diffusivity has a marginal impact on thecomputational results. To assess the extent to which numericaldiffusion influences the outcomes of our simulations, grideffects were assessed, as discussed later in the article.The scalar concentration field c is two-way-coupled to the

flow field; that is, c is convected by the flow (see ui in eq 1), andat the same time, it affects the flow because c determines thelocal viscosity and density of the liquid mixture. For viscosity aswell as density, linear dependencies in c are assumed: νmx = νD− c(νD − νL) and ρmx = ρ̅ + (1/2 − c)Δρ, where νmx and ρmx arethe local mixture viscosity and density, respectively. The lattice-Boltzmann method is able to directly deal with the place- andtime-varying viscosity. The density is incorporated through a

Figure 2. Liquid composition c in a vertical, midbaffle plane at (from top to bottom) 10, 20, 30, and 50 impeller revolutions after startup. From leftto right, Ri = 0.0, 0.125, and 0.5. All cases: Re = 6000 and Vi = 1.0. Down-pumping impeller at clearance C = T/3.

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Boussinesq approximation: A liquid element having density ρmxfeels a body force in the positive z direction (as defined inFigure 1) of fz = g(ρ̅ − ρmx) = gΔρ(c − 1/2), and this force isincorporated into the LB scheme. In the Boussinesqapproximation, the body force term is the only place wherethe density variation enters the Navier−Stokes equations. Forthis approximation to be valid, it must be true that Δρ/ρ̅ ≪ 1[that is, Ri ≪ g/(N2D)].Numerical Settings. The default grid (which, as explained

above, is uniform and cubic) has a spacing Δ such that 180Δcorresponds to the tank diameter T. The number of time stepsto complete one impeller revolution is 2000. As a result, the tipspeed of the impeller is πND = 0.094 in lattice units (with animpeller diameter of D = T/3), which keeps the flow velocitiesin the tank well below the speed of sound of the lattice-Boltzmann system, thus achieving incompressible flow. Toassess grid effects (related to the flow as well to scalartransport), a number of simulations were performed on finergrids: T = 240Δ, 330Δ, 420Δ, and 552Δ. Because of theexplicit nature of the lattice-Boltzmann method and its(in)compressibility constraints, the finer grids required moretime steps per impeller revolution (up to 6400 for the finestgrid).The microscale of turbulence (Kolmogorov length scale η)

relates to a macroscopic length scale (say, the tank diameter T)according to η = T × Re−3/4. The criterion for sufficientlyresolved direct numerical simulations of turbulence is Δ <πη.21,22 According to this criterion, at Re = 6000, a grid with T= 180Δ slightly under-resolves the flow; at our highestReynolds number (Re = 12000), the same grid has Δ ≈ 6.4η,and the finest grid (with T = 552Δ) has Δ ≈ 2.1η. As discussedabove, full resolution of the Batchelor scale (ηB) is not anoption, as it is a factor of 30 smaller than the Kolmogorov scale,so that, on the finest grid and at Re = 12000, Δ ≈ 60ηB. Theconsequences of not resolving ηB were assessed through gridrefinement.The relatively modest default resolution of T = 180Δ was

chosen to limit computational cost and, at the same time, toallow for a significant number of simulations with sometimeslong time spans (up to 300 impeller revolutions) required toreach sufficient levels of homogeneity so that mixing timescould be determined. The large number of cases is the result ofparameter variations: Re, Ri, Vi, off-bottom impeller clearance,and direction of rotation of the impeller were all varied.

■ RESULTSFlow and Scalar Field Impressions for a Down-

Pumping Impeller at C = T/3. The results of our simulationsare discussed mostly in terms of the flow and concentrationfields in the vertical, midbaffle cross section as they evolve intime from startup from a zero-flow, fully segregated, stablestate. The set of base cases that begins this section focuses onthe effect of a density difference (and thus of Ri) on mixing. Ithas Re = 6000, Vi = 1.0, and an impeller at C = T/3 that pumpsthe liquid downward. Figure 2 shows the scalar concentrationfields for three different values of Ri and at different momentsafter startup of the impeller. One of cases is nonbuoyant andthus a passive-scalar case (Ri = 0.0 and Vi = 1.0). Buoyancyclearly impacts the mixing process. At Ri = 0.0, the interfacebetween high and low concentration quickly disintegrates, andlow-concentration blobs appear in the high-concentrationupper portion of the cross section as a result of three-dimensional flow effects, largely because of the presence of

baffles. At Ri = 0.125, this is less the case. The interface isclearly agitated but largely maintains its integrity (i.e., it is notbroken up). At still larger Ri (Ri = 0.5 in Figure 2), the interfacestays more or less horizontal. It rises as a result of erosion:High-concentration (and thus low-density) liquid is erodedfrom the interface and drawn down to the impeller. It thenquickly mixes in the lower part of the tank. This leads to agradual rise of concentration in the part of the tank underneaththe interface. The portion above the interface stays at the initialconcentration and gradually reduces in height. The interfacealso acts as a barrier for momentum transfer: Above theinterface, there is hardly any flow, and the flow that is there ismainly generated by the revolving shaft that extends into thatregion, not so much by the action of the impeller (see Figure 3,which has a logarithmic color scale).

Compared to the impact of the Richardson number, theeffect of the Reynolds number is relatively modest, as can beassessed from Figure 4, in which we compare, at Ri = 0.125 andVi = 1.0, the scalar concentration fields in the midbaffle plane atRe = 3000 and Re = 12000 at two moments in time. Thesefields have their Re = 6000 counterparts displayed in Figure 2(middle column, second and fourth panels from the top). At allthree Reynolds numbers, the interface reaches a level of z ≈2.2D after 20 revolutions and is close under the lid after 50revolutions.A viscosity ratio larger than 1 has an interesting effect on the

mixing process: See Figure 5, where results for the highestviscosity ratio (Vi = 4.0) are shown. Because the upper liquid isnow less viscous, it is easier for momentum to penetrate theupper parts of the tank volume; the interface between denseand light liquids gets more agitated compared to correspondingsituations for Vi = 1.0 (see Figure 2). At the same time, thehigher viscosity in the lower part of the tank slowshomogenization there (again compare Figure 5 with corre-sponding panels of Figure 2). The net result of these opposingeffects on mixing time is assessed in the next section.

Mixing Time Analysis. Mixing times were analyzed basedon the scalar concentration fields in the midbaffle, vertical crosssections through the tank. In these cross sections, wemonitored the spatial concentration standard deviation as afunction of time

Figure 3. Instantaneous realization 50 revolutions after startup of thevelocity magnitude in a midbaffle plane for Ri = 0.5, Vi = 1.0, and Re =6000 and a down-pumping impeller at C = T/3. Note the logarithmiccolor scale.

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∬σ = = − ⟨ ⟩tA

c x y z t c t x z( )1

{ ( , 0, , ) [ ( )] }d dA

2 2 2

where ⟨c⟩(t) is the average scalar concentration in the midbaffleplane at moment t. At time zero, the liquids are fully segregated,with c = 1 in the upper half of the cross section, c = 0 in thelower half, and ⟨c⟩ = 0.5, so that the starting value of thestandard deviation is σ(t = 0) = 0.5.For a down-pumping impeller at clearance C = T/3, the time

series of standard deviations are given in Figure 6 for Re =

6000, Vi = 1.0, and various Richardson numbers. The decayrates strongly depend on the Richardson number; at Ri =0.03125 the scalar variance decay is close to that of a passivescalar, so one can conclude that (for the specific stirred-tankconfiguration and process conditions) buoyancy influences thehomogenization process if Ri ≥ 0.0625. To characterize thedecay of scalar variance with a single number, the time to reachσ = 0.025 was chosen here as the mixing time measure τσ.Figure 7 shows how the dimensionless mixing time τσNdepends on Ri and Re for Vi = 1.0.

As discussed in our previous article,1 for Ri = 0.0 and Vi = 1.0(i.e., no buoyancy and no viscosity variation), the above-definedmeasure τσ agrees with correlations for mixing times (symbolτM) based on experimental data: Grenville et al.23 suggestedτMN = 5.1(Po)−1/3(T/D)2, where Po is the power number[which is the power P drawn by the impeller made

Figure 4. Assessment of Reynolds number effects at Ri = 0.125 and Vi= 1.0 for a down-pumping impeller at C = T/3. (Left) Re = 3000,(right) Re = 12000. (Top) 20 and (bottom) 50 impeller revolutionsafter startup.

Figure 5. Assessment of viscosity ratio effects at Re = 6000 with adown-pumping impeller at C = T/3. (Left) Ri = 0.125, Vi = 4.0;(right) Ri = 0.5, Vi = 4.0. (Top) 20 and (bottom) 50 impellerrevolutions after startup. Compare with the corresponding panels inFigure 2 for Vi = 1.0.

Figure 6. Scalar standard deviation in the midbaffle plane (σ) as afunction of time; comparison of different Richardson numbers for adown-pumping impeller at clearance C = T/3, Re = 6000, and Vi = 1.0.The dashed curve has Ri = 0.0. The solid curves have Ri = 0.03125,0.0625, 0.125, 0.25, and 0.5 in the order indicated.

Figure 7. Mixing time based on scalar standard deviation (τσ) versusRi at three different Reynolds numbers. Down-pumping impeller at C= T/3, Vi = 1.0.

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dimensionless according to Po ≡ P/(ρD5N3)]. The mixing timedata in Figure 7 for Ri = 0.0 are τσN = 47.8, 40.1, and 36.1 forRe = 3000, 6000, and 12000, respectively. For a four-blade, 45°down-pumping pitched-blade turbine with C = T/3 and T/D =3, Chapple et al.24 reported Po in the range of 1.1−1.2 for 103 ≤Re ≤ 104. The power numbers derived from our simulations areslightly higher: Po = 1.37, 1.35, and 1.34 for Re = 3000, 6000,and 12000, respectively. The mixing time correlation then givesτMN in the range of 41−44, which is close to the results for τσNpresented in Figure 7 for Ri = 0.0 and Vi = 1.0. To some extent,this agreement justifies our choice for taking σ = 0.025 as theuniformity criterion.Summarizing the mixing process in the single number τσN is

a strong simplification, however. In Figure 8, the concentrationfields at a moment just after the level σ = 0.025 has beenreached are given for the simulations of Figure 6 with nonzeroRi. Note that, in this figure, the color scale is enhanced toemphasize deviations from the (ultimate) c = 0.5 concentration.The states of mixing for the different Richardson numbers areclearly different, although σ ≈ 0.025 in all cases. For high Ri, theconcentration throughout the tank is fairly uniform except for afew patches with high c near the lid. For lower Ri, there is more(though weaker) variation in c throughout the tank. Despitethese observations, in the following discussion, the results ofour simulations are summarized mainly in terms of τσN.Figure 7 shows how, in the case of uniform kinematic

viscosity (Vi = 1.0), the dimensionless mixing time τσN relatesto Ri and Re: the higher Ri, the larger the mixing time; thelarger Re, the lower the mixing time. Buoyancy amplifies thedifferences in mixing times between the various Reynoldsnumbers: At Ri = 0.0, the mixing times of the three Reynoldsnumbers are within 25%. By Ri = 0.0625, this has grown tosome 50%, and differences increase further for higher Ri.Specifically, homogenization at the lowest Reynolds number(3000) slows dramatically. At this Re value, the stratificationmakes it hard to sustain turbulence, specifically in the higherlevels of the tank. Reduced fluctuation levels also have anegative impact on the interface erosion process that, asdiscussed above, becomes increasingly rate-determining athigher Richardson numbers.The viscosity ratio affects the mixing time only weakly; see

Figure 9. If there is any discernible trend in this figure, it is aslight increase in mixing time with Vi. As demonstrated in thediscussion of the effects of spatial resolution later in this article,however, the differences in Figure 9 are of the same order ofmagnitude as the differences between different grids and aretherefore considered not to be significant.Impeller Rotational Direction and Off-Bottom Clear-

ance. As is clear from Figure 7, the mixing time is a strongfunction of the Richardson number, with much longer mixingtimes for higher Ri values. This is sufficient reason to investigate

whether simple/minor design changes can accelerate theblending process. As a first simple change, the rotationdirection of the impeller was reversed so that it pumped liquidupward (i.e., in the direction against gravity). This implies thatthe denser liquid was directly pumped into the upper part ofthe tank that contained lighter liquid. It also implies that theinterface between light and dense liquid became much moreagitated. The consequences for the mixing process for arelatively high Richardson number of 0.5 (and Re = 6000 andVi = 1.0) can be observed in Figure 10 (left panels). Reversingthe impeller helped in dramatically decreasing the mixing timeas compared to that obtained with the down-pumping impeller(see Figure 11, left panel). For Ri ≥ 0.125 a reduction of τσN bya factor of 2 or more was achieved. As can be seen in Figure 10(lower left panel), in the final stages of mixing, a layer of lightliquid persisted in the top part of the tank that became rate-limiting for the mixing process.As a possible solution, we considered placing the (still) up-

pumping impeller higher in the tank at C = T/2. This indeedremoved the lighter liquid from the top of the tank morequickly but instead left denser liquid at the bottom (middlepanels of Figure 10). In terms of dimensionless mixing time,the up-pumping impeller at C = T/2 performed equally well asthe up-pumping impeller at C = T/3 (see the right panel ofFigure 11).To finish this small exercise in tank-layout variation, the case

with C = T/2 and a down-pumping impeller was simulated aswell; see the right panels of Figure 10. This layout also leftsome light liquid at the top of the tank for long times. Theagitator placed at the same level, z = H/2, as the interface verymuch helped in bringing the mixing time down to levelscomparable to those obtained for C = T/2 and C = T/3 with anup-pumping impeller, as can be witnessed from comparing thedata in the two panels in Figure 11.

Figure 8. Concentration fields in the midbaffle plane the moment just after the σ = 0.025 level was reached for the cases shown in Figure 6 (exceptthe Ri = 0 case). From left to right: Ri = 0.03125, 0.0625, 0.125, 0.25, and 0.5. Note that the color scaling is finer than in Figures 2 and 4.

Figure 9. Mixing time based on scalar standard deviation (τσ) versusRi at three different viscosity ratios. Down-pumping impeller at C = T/3, Re = 6000.

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The results presented so far in this subsection are for Vi =1.0. As was previously shown in Figure 9 for the default system(C = T/3, down-pumping), an increase in viscosity ratio wasfound to have a minor effect on mixing time for the alternativelayouts as well. As an example, we present mixing times for anup-pumping impeller at C = T/3 in Figure 12.

Assessment of Resolution Effects. The resolution of theflow dynamics and, even more so, the resolution of the scalartransport were marginal (to say the least) for the default gridwith a spacing of Δ = T/180. This resolution was chosen forpractical reasons: it allowed us to complete many simulationsover many impeller revolutions per simulation. Resolutionissues are particularly critical for high Reynolds numbers andalso for high viscosity ratios. The former is true because higherReynolds numbers have a wider spectrum of length scales; thelatter because blending a high-viscosity liquid into a low-

Figure 10. Concentration contours in the midbaffle plane at Re = 6000, Vi = 1, and Ri = 0.5 after (from top to bottom) 10, 20, 30, and 50 impellerrevolutions. (Left) Up-pumping impeller at C = T/3, (middle) up-pumping impeller at C = T/2, and (right) down-pumping impeller at C = T/2.

Figure 11. Mixing time based on scalar standard deviation (τσ) versusRi at Re = 6000 and Vi = 1.0. (Left) Comparison between down- andup-pumping impellers at C = T/3. (Right) Cases with C = T/2compared with (the default) down-pumping impeller at C = T/3.

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viscosity liquid might go through a stage with thin striations ofone liquid phase inside the other. Such striations would need tobe resolved in a direct simulation. With these points in mind,two cases with the (default) down-pumping impeller at C = T/3 were selected for an assessment of grid effects: (1) a case withRe = 12000 and further Ri = 0.0625 and Vi = 1.0 and (2) a casewith Vi = 4.0, Re = 6000, and Ri = 0.0. Five grids werecompared for case 1, three for case 2. The grids were mainly

compared in terms of scalar transport results (concentrationfields, decay of scalar variance, and mixing time) becauseresolution requirements are higher for scalar transport than forflow dynamics and because flow dynamics and scalar transportare tightly coupled: It is not likely that a poor flow fieldprovides a good scalar field.Figure 13 shows a qualitative comparison of the scalar field

for the high-Re case on the five grids at the same moment intime (20 revolutions after startup). Although the overall levelsof mixing are comparable between the grids, clearly much moredetail is captured by the finer grids, with thin structures visiblein the impeller outstream for Δ = T/552 that are not there forΔ = T/180. To some extent, this observation is reflected in thedecay of scalar variance, as sown in Figure 14. The decays up totN ≈ 15 are very similar for the different grids. Beyond 15revolutions, the decays are slightly slower for the finer grids.This implies that the macroscopic transport is adequatelycaptured by the coarser grid. Once the scalar length scalesbecome finer, diffusion sets in, more quickly so for the coarsergrids. At least in terms of the scalar variance, this effect does notlead to very large differences between the grids. The mixingtime based on the σ = 0.025 criterion differs by some 10%, at amaximum, with τσN highest for the finest grid (Figure 14). This10% we consider (a lower bound for) the accuracy of ourmixing time results.

Figure 12. Mixing time based on scalar standard deviation (τσ) versusRi at Re = 6000 with an up-pumping impeller at C = T/3. Comparisonbetween three different viscosity ratios.

Figure 13. Instantaneous realizations (20 impeller revolutions after startup) of the scalar concentration in the midbaffle plane for Re = 12000, Ri =0.0625, Vi = 1.0 with a down-pumping impeller at C = T/3. Qualitative comparison between different spatial resolutions: Δ = T/180, Δ = T/240, Δ= T/330, Δ = T/420, Δ = T/552. The size of each panel scales linearly with the grid size.

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Grid effects for the high-viscosity-ratio case (Vi = 4.0) arepresented in a similar fashion in Figure 15. Again, a typical

accuracy of 10% can be concluded. The trend in mixing timewith respect to resolution is opposite: higher resolution nowgives lower τσN. Inspection of concentration fields (not shownfor brevity) indicates that the entrainment of the low-viscosityliquid in the high-viscosity liquid by the impeller is associatedwith thin layers of low-viscosity material. These layers are betterresolved by the finer grid, (apparently) resulting in fastermixing.

■ SUMMARY, CONCLUSIONS, AND OUTLOOKIn this work, homogenization of two initially segregatedNewtonian liquids with different physical properties (densityand viscosity) in stirred tanks was simulated. The tanks werecylindrical, baffled, and flat-bottomed; the impeller was a four-blade, 45° pitched-blade turbine. Given the tank and impellerlayout, the blending process is governed by three dimensionlessnumbers: a Reynolds number (Re), a Richardson number (Ri),and a viscosity ratio (Vi). As a fourth dimensionless parameter,the Schmidt number (Sc), was also identified. It relates to thediffusivity of the two miscible liquids in one another. Thisdiffusivity was assumed to be small, so that Sc was large. In thislimit of large Sc, given the resolution limitations of thesimulations, it was not possible to investigate the effects of Scon the blending process.The Reynolds numbers were in the range of 3000−12000. In

this range, indicating transitional and moderately turbulentflow, we did not need to use turbulence modeling to simulate

the flow. This was a deliberate choice, as it removesuncertainties as to how the turbulence modeling deals withthe local and temporal density and viscosity variations. In ourdirect simulations, these variations acted on the (resolved)Navier−Stokes equations directly and in a physically consistentmanner. At various levels of the simulations, spatial resolutionwas a concern: (1) Direct simulation of turbulence requiresresolution of the flow down to the Kolmogorov length scale.(2) Scalar transport at high Sc values makes the Batchelor scaleeven smaller. (3) Liquids with different viscosities are known topenetrate one another through thin layers (striations).Resolution was handled in a pragmatic way: through checkingthe effects of refining the grid on the blending process. The gridspacing was refined up to a factor of 3 from the default grid. Inthe explicit, three-dimensional simulation procedure based onthe lattice-Boltzmann method, such a refinement implies anincrease in computational effort of 34 = 81 (the power 4 is thesum of 3 spatial dimensions plus one time dimension). Mixingtime variations of 10% between the grids were observed, andthe coarsest grid (with 180 grid spacings along the tankdiameter) was deemed a fair compromise of accuracy on oneside and modest computational demands on the other; itallowed us to run many cases for many impeller revolutions.With the impeller in down-pumping mode, placed one-third

of the tank diameter above the bottom of the tank, and startingfrom a stable stratification, the Richardson number was foundto have the most impact on the blending process: the mixingtime steeply increases with increasing Ri. For high Ri (on theorder of 0.5), blending is largely due to erosion of the interfacebetween dense liquid and light liquid. Erosion is brought aboutby the agitation of the denser liquid underneath the interface;the lighter liquid near the top hardly feels the action of theimpeller. At the other side of the Ri spectrum, buoyancy effectsbecome insignificant for Ri on the order of 0.03. The increaseof mixing time with Ri is more pronounced at lower Reynoldsnumbers than for higher Re. The viscosity ratio has a weakimpact on the mixing time, at least in the viscosity ratio range(1 ≤ Vi ≤ 4) investigated in this work.Simple geometrical measures can bring down the mixing

time significantly: Changing from down- to up-pumping modedecreases the mixing time by a factor of sometimes more than2. The same holds true for changing the vertical location of theimpeller. The key issue in decreasing the mixing time for bothdesign measures is more direct agitation of the interfacebetween the two liquids.This purely computational study asks for experimental

guidance and (possibly/hopefully) validation. Specifically, thenumerical issues regarding resolution require experimentalassessment. It should be noted, however, that experiments alsoface resolution questions. For instance, in visualizationexperiments that would monitor the decay of scalar varianceas a metric for mixing (analogously to what we did simulation-wise), the scalar variance is to some extent a function of theresolution and dynamic range of the recording device (i.e., thecamera).

■ AUTHOR INFORMATION

Corresponding Author*E-mail: jos@ualberta.ca.

NotesThe authors declare no competing financial interest.

Figure 14. (Left) Scalar standard deviation in the midbaffle plane (σ)as a function of time; comparison at different resolutions for the casesdisplayed in Figure 13, τσN. (Right) Mixing time as a function ofresolution, where G is the number of grid spacings per tank diameter T(e.g., for Δ = T/180, G = 180).

Figure 15. (Left) Scalar standard deviation in the midbaffle plane (σ)as a function of time; comparison at different resolutions for Re =6000, Ri = 0.0, and Vi = 4.0 for a down-pumping impeller at C = T/3.(Right) Mixing time as a function of resolution.

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